classifying spaces for proper actions of locally finite groups

CLASSIFYING SPACES FOR PROPER 

ACTIONS OF LOCALLY FINITE GROUPS 

Warren Dicks, Peter H. Kropholler, Ian J. Leary and Simon Thomas 

April 16, 2002 

For each finite ordinal n, and each locally finite group G of cardinality ℵ n ,weconstruct 

an (n + 1)-dimensional, contractible CW-complex on which G acts with finite 

stabilizers. We use the complex to obtain information about cohomology with induced 

coefficients. Our techniques also give information about the location of some 

large free abelian groups in the hierarchy H . 

Throughout, let G be a group, and let A be a Z-module with trivial G-action. 

We let AG denote the induced ZG-module ZG ⊗ Z A. Ring-actions on modules 

and group-actions on sets are tacitly understood to be on the left, if not specified 

otherwise. 

1. Holt’s Conjectures 

The purpose of this section is to describe our main algebraic results, Theorems 

3.10 and 5.4, and place them in the context of prior work. 

1.1 Notation. Let rank(G) denote the smallest cardinal κ such that there exists 

some set of generators of G of cardinality κ. 

If G is not finitely generated, then rank(G) =|G|, and we define ℵ-rank(G) tobe 

the ordinal α such that rank(G) =ℵ α ;ifG is finitely generated, then rank(G) < |G|, 

and we set ℵ-rank(G) =−1. 

Recall that, for each ordinal α, ω α denotes the least ordinal of cardinality ℵ α . 

We find it convenient to set ℵ −1 =1. 

Throughout this section, let n ∈ N (= ω 0 ). □ 

D. F. Holt proposed the following description of the cohomology with induced 

coefficients, for locally finite groups. 

1.2 Conjecture (Holt [6]). If G is locally finite, then 

| H n (G, AG)| = |A| ℵ n−1 

if n = ℵ-rank(G)+1, 

H n (G, AG) =0 if n ≠ ℵ-rank(G)+1. 

Commentary. In Examples 3.3 we recall that, for any group G, 

{ A if G is finite and n =0, 

(1.3) H n (G, AG) = 

0 if G is finite or n = 0, but not both. 

1991 Mathematics Subject Classification. Primary 20F50; Secondary 20J06, 20K20, 57M07. 

1

2 DICKS, KROPHOLLER, LEARY AND THOMAS 

Thus the conjecture really concerns the cases where n ≥ 1andG is infinite, and 

the notation has been artificially contrived to embrace the trivial marginal cases. 

For any infinite group G, thesetofcocyclesforG with coefficients in AG is of 

cardinality |A| |G| ,so1≤|H n (G, AG)| ≤|A| |G| . If G is infinite and locally finite, 

then the conjecture implies that only the extreme values can be achieved. □ 

We now briefly state the cases which are known, including those obtained in this 

paper. 

1.4 Notation. We say that G has the finite extension property for proper subgroups 

if each proper subgroup of G is a proper subgroup of finite index in some subgroup 

of G. For example, abelian torsion groups have this property. 

Let us say that A is o(G)-inverting if, for every finite subgroup H of G, multiplication 

by |H| gives an automorphism of A; equivalently, for each g ∈ G whose 

order o(g) is finite, multiplication by o(g) gives an automorphism of A. 

If R is a ring (associative, with 1), then R is o(G)-inverting, as Z-module, if and 

only if the order of each finite subgroup of G is a unit in R. If R is not o(G)-inverting, 

then it is easy to show that cd R G, the cohomological dimension of G with 

respect to R, is∞, a value which we shall think of as ω 0 = ℵ 0 . □ 

1.5 Known cases of Conjecture 1.2. Let G be a locally finite group. 

(1) H n (G, AG) =0if n>ℵ-rank(G)+1. 

(2) If G has the finite extension property for proper subgroups, then 

H n (G, AG) =0if n ≠ ℵ-rank(G)+1. 

(3) For n ∈{0, 1}, H n (G, AG) =0if n ≠ ℵ-rank(G)+1. 

(4) For n ∈{0, 1, 2}, | H n (G, AG)| = |A| ℵ n−1 

if n = ℵ-rank(G)+1. 

(5) It is consistent with ZFC that 

| H n (G, AG)| ≥2 ℵ n−1 

if n = ℵ-rank(G)+1 and A is nonzero. 

Hence, it is consistent with ZFC that 

| H n (G, AG)| = |A| ℵ n−1 

if n = ℵ-rank(G)+1 and |A| ≤ℵ n−1 . 

Commentary. (1), the “easy” part of Conjecture 1.2, is proved in Theorem 3.10. It 

was proved in [11] for the case where A is o(G)-inverting, and, before that, in [4], [5] 

for the case where A is o(G)-inverting and torsion. 

(2) was proved by Holt [5]. We give another proof of the abelian case in Corollary 

6.10. 

(3). By (1.3), this holds for n = 0. It was proved by Holt [6] for n = 1; 

see Theorem 6.4. 

(4). By (1.3), this holds for n =0. Itiswellknownforn = 1; see Theorem 4.5. 

In Theorem 5.4, we prove it for n = 2; Holt [6] had previously shown this was 

consistent with ZFC, see [14, Section 1]. 

(5). Suppose that n = ℵ-rank(G)+1andthatA is nonzero. 

We shall now see that it is consistent with ZFC that | H n (G, AG)| ≥2 ℵ n−1 

. 

For each prime p, wewriteZ(p ∞ ) := 

lim 

m→∞ Z/pm Z,where,form ∈ N, themap 

Z/p m Z → Z/p m+1 Z is given by multiplication by p. 

We claim that there exists a Z-module k, andaZ-submodule A ′ of A, such that 

the quotient A/A ′ is isomorphic to k, and either k = Q, or there exists a prime p 

such that k = Z/pZ or k = Z(p ∞ ).

CLASSIFYING SPACES FOR PROPER ACTIONS OF LOCALLY FINITE GROUPS 3 

Consider first the case where A is not divisible, so there exists a prime p such 

that A/pA is nonzero. But A/pA is a direct sum of Z-submodules each of which is 

isomorphic to Z/pZ. Hence A/pA projects onto any such summand. 

If A is divisible, then A is a direct sum of Z-submodules each of which is isomorphic 

to Q or to Z(p ∞ )forsomeprimep; see, for example, [3, Theorem IV.23.1]. 

Hence A projects onto any such summand. 

In all cases, we can find A ′ , k as claimed. 

Now there is a long exact sequence in cohomology which contains the subsequence 

H n (G, AG) → H n (G, kG) → H n+1 (G, A ′ G). 

By (1), H n+1 (G, A ′ G)=0,so| H n (G, AG)| ≥|H n (G, kG)|. 

Thus, for the first part, it remains to show that it is consistent with ZFC that 

| H n (G, kG)| ≥2 ℵ n−1 

. 

It is proved in [11] that, if k is an o(G)-inverting prime field, then it is consistent 

with ZFC that | H n (G, kG)| ≥2 ℵ n−1 

. However on carefully reading that proof, 

one sees that all applications of the o(G)-inverting hypothesis can be replaced with 

applications of (1), so, in fact, it is proved that it is consistent with ZFC that if k 

is a prime field then | H n (G, kG)| ≥2 ℵ n−1 

. 

It remains to consider the case where k = Z(p ∞ )forsomeprimep. Again, 

it is not difficult to show that the argument in [11] can be further modified to 

cover this case by interpreting dim k m := m for all m ∈ N. If M is a finitely 

generated (free) Z-submodule of ZG, andkM denotes the image of the natural 

map k ⊗ Z M → k ⊗ Z ZG = kG, then one can show dim kM =rankM, sincek is 

divisible. Moreover, if M ′ is a Z-submodule of M, thendimkM ′ ≤ dim kM, and, 

if equality holds, then kM ′ = kM, sincek is divisible. Using these observations, 

one can verify that the argument in [11] applies with k = Z(p ∞ ). 

It follows that, in all cases, it is consistent with ZFC that | H n (G, kG)| ≥2 ℵ n−1 

. 

Now suppose that |A| ≤ℵ n−1 . Hence |A| ℵ n−1 

=2 ℵ n−1 

; see [7, p.49] for the case 

where n ≥ 1. Thus, it is consistent with ZFC that | H n (G, AG)| ≥|A| ℵ n−1 

. 

We previously observed that | H n (G, AG)| ≤|A| ℵ n−1 

, so it is consistent with 

ZFC that | H n (G, AG)| = |A| ℵ n−1 

. 

This proves (5). It had previously been proved by Holt [5] in the case where G 

has the finite extension property for proper subgroups; see [14, Section 1]. □ 

We wish to refine part of Conjecture 1.2. 

1.6 Conjecture. If G is locally finite, and n = ℵ-rank(G) +1,thenH n (G, AG) 

contains a Z-submodule isomorphic to A ℵ n−1 

, and hence | H n (G, AG)| = |A| ℵ n−1 

. 

Commentary. By (1.3), this holds for n = 0. It is probably well known for n =1; 

see Theorem 4.5. In Theorem 5.4, we prove it for n =2. □ 

Conjecture 1.2 was preceded by, and motivated by, an earlier proposal, concerning 

the cohomological dimension of locally finite groups.


1.7 Conjecture. If G is a locally finite group, and R is a nonzero, o(G)-inverting 

ring, then cd R G =min{ℵ-rank(G)+1, ∞}. 

Commentary. Holt [4] proposed this conjecture with the additional hypothesis that 

R is a field of prime order, and, in [11], the additional hypothesis was weakened to 

R being commutative. 

Notice that min{ℵ-rank(G)+1, ∞} can be expressed as inf{n ∈ N |ℵ n > |G|}, 

where the infimum of the empty set is taken to be ∞. 

The inequality cd R G ≤ℵ-rank(G) + 1 follows from a classic result of Auslander 

[1, Proposition 3]; see [12, Lemma 3.7] or Theorem 3.10 below. 

Cohomological dimension cannot increase on passing to a subgroup, so we may 

assume that ℵ-rank(G)


We let F denote the class of finite groups. Notice that F ∩ sub(G) is a subgroup-closed 

G-family. A space of type E(G, F) is called an EG. (It is also called a 

classifying space for proper G-actions, that is, G-actions with finite stabilizers.) □ 

The following is well known. 

2.2 Proposition. If X is a subgroup-closed G-family, then there exists a space of 

type E(G, X), andanyG-map between two spaces of type E(G, X) is a G-homotopy 

equivalence. 

Proof. The first part can be seen by Milnor’s construction. Thus, let ∆ be any 

G-set such that X is precisely the set of subgroups of G which fix at least one point 

of ∆. Let X =∆∗ ∆ ∗ ∆ ∗···, the union of iterated joins of ∆. Then X is a space 

of type E(G, X). 

For the second part, see, for example, [15, Proposition II.2.7]. □ 

2.3 Corollary. If X 1 ⊆ X 2 are subgroup-closed G-families, and X 1 (resp. X 2 ) 

is a space of type E(G, X 1 )(resp. E(G, X 2 )), then there exists a cellular G-map 

X 1 → X 2 . 

Proof. The join X 1 ∗ X 2 is a space of type E(G, X 2 ), and the inclusions 

ι 1 : X 1 → X 1 ∗ X 2 , ι 2 : X 2 → X 1 ∗ X 2 

are G-maps. By Proposition 2.2, ι 2 is a G-homotopy equivalence, and the homotopy 

inverse X 1 ∗ X 2 → X 2 composed with ι 1 gives a G-map X 1 → X 2 . This 

is then G-homotopic to a cellular G-map X 1 → X 2 ; see, for example, [15, Theorem 

II.2.1]. □ 

One could give a dual proof, using the projection maps from the Cartesian product 

X 1 × X 2 , which is a space of type E(G, X 1 ). 

The following is a topological analogue of a classic result of Auslander [1, Proposition 

3]. 

2.4 Theorem. Let β be a limit ordinal, let (G α | α ≤ β) be a continuous chain of 

subgroups of G, andlet(X α | α ≤ β) be a continuous chain of subsets of sub(G) 

such that, for each α ≤ β, X α is a subgroup-closed G α -family. 

Let n ∈ N, and suppose that, for each α


By Proposition 2.2, since X α and Y α are of type E(G α , X α ) there exists a cellular 

G α -map Y α → X α , and hence a cellular G α+1 -map G α+1 × Gα Y α → G α+1 × Gα X α . 

Take X α+1 to be the identification space, or pushout, 

G α+1 × Gα Y α −−−−→ M α 

⏐ 

⏐ 

↓ 

↓ 

G α+1 × Gα X α −−−−→ X α+1 . 

Notice that dim X α+1 = n +1, since dimY α = n, dimM α = n +1, and 

dim X α ≤ n + 1. It is not difficult to check that X α+1 is of type E(G α+1 , X α+1 ). 

This completes the proof. □ 

2.5 Remark. For n =0andβ = ω 0 , the construction in the above proof gives the 

Bass-Serre tree of the graph of groups corresponding to the countable ascending 

chain (G α | α


2.8 Theorem. If n ∈ N, andℵ-rank(G)


3.3 Examples. Let G be a finite group. 

Here, AG = A[[G]], so induced ZG-modules are co-induced, and hence G-acyclic. 

Suppose that M is an o(G)-inverting ZG-module. Then the multiplication map 

M[[G]] → M, 

∑ 

g∈G 

m g .g ↦→ ∑ g∈G 

gm g , 

is ZG-split with right inverse m ↦→ 1 ∑ 

g −1 m.g. Here, M is a ZG-summand of 

|G| 

g∈G 

an induced ZG-module, so M is G-acyclic. □ 

In the following, G acts on tensor products over Z via the diagonal action. 

3.4 Lemma. Let M be a ZG-module. 

(1) The functor Hom ZG (−⊗ Z ZG, M) carries Z-split exact sequences of ZG-modules 

to exact sequences. 

(2) Let H be a subgroup of G. IfM is H-acyclic, then the functor 

Hom ZG (Z[G/H] ⊗ Z −,M) 

carries augmented ZG-projective resolutions of Z to exact sequences. 

Proof. (2). Let L be a ZG-module. There is a natural identification of ZG-modules, 

with gH ⊗ l corresponding to g ⊗ g −1 l. 

It follows that we can identify 

Z[G/H] ⊗ Z L = ZG ⊗ ZH L, 

Hom ZG (Z[G/H] ⊗ Z −,M)=Hom ZG (ZG ⊗ ZH −,M)=Hom ZH (−,M) 

as functors on ZG-modules. Since M is H-acyclic, this functor carries augmented 

ZG-projective resolutions of Z to exact sequences. 

(1) is proved similarly. □ 

3.5 Notation. Let X be a G-complex. 

We shall treat X as the G-set whose elements are the open cells of X. The cellular 

chain complex of X is then the permutation module ZX, with the structure of a 

differential graded ZG-module, with differential ∂ of degree −1. Here the grading 

is that determined by the dimensions of the cells, so the nth component C n (ZX) 

has as Z-basis the cells of dimension n. 

We let η: X × X → Z denote the function such that ∂x = ∑ η(x, y).y for each 

x ∈ X. Thus,ifx is an n-cell, then η(x, y) = 0 unless y is one of the finitely many 

(n − 1)-cells incident to x. □ 

The following is a degenerate case of the equivariant cohomology spectral sequence; 

see, for example, [2, VII.7.10(7.10)]. 

y∈X


3.6 Theorem. Let X be an acyclic G-complex. If M is a ZG-module which is 

G x -acyclic for each x ∈ X, then H ∗ (Hom ZG (ZX, M)) ≃ H ∗ (G, M), as graded 

abelian groups. 

Recall that Hom ZG (ZX, M) denotes the differential graded abelian group with 

nth component C n (Hom ZG (ZX, M)) = Hom ZG (C n (ZX),M). 

Proof. The homology of (ZX, ∂) isZ, concentrated in degree zero. 

We choose a free ZG-resolution of Z, andwriteitas(ZY,∂) forsomeG-free 

G-set Y ; for example, we could take Y to be an E G, andZY its cellular chain 

complex. Then Hom ZG (ZY,M) is an additive abelian differential graded group, 

and its cohomology is H ∗ (G, M). 

We consider the double complex ZX ⊗ Z ZY with diagonal G-action, and the 

double complex Hom ZG (ZX ⊗ Z ZY,M). We get a fourth-quadrant commuting 

diagram which can be schematically represented as 

0 

⏐ 

↓ 

(3.7) 

Hom ZG (ZX, M) 

⏐ 

↓ 

0 −−−−→ Hom ZG (ZY,M) −−−−→ Hom ZG (ZX ⊗ Z ZY,M). 

To show that the cohomology group of the outer row, Hom ZG (ZX, M), is isomorphic 

to the cohomology group of the outer column, Hom ZG (ZY,M), it suffices 

to show that the remaining, or inner, rows and columns of (3.7) are exact. Each 

inner column is exact because Hom ZG (ZX ⊗ Z −,M) is exact on augmented projective 

ZG-resolutions of Z, by Lemma 3.4(2). Similarly, each inner row is exact 

because Hom ZG (−⊗ Z ZY,M) isexactonZ-split exact sequences of ZG-modules, 

by Lemma 3.4(1). □ 

3.8 Corollary. Let M be a ZG-module, and let X be a finite-dimensional acyclic 

G-complex. If M is G x -acyclic for each x ∈ X, then H n (G, M) = 0 for all 

n>dim X. □ 

We record the case of finite stabilizers. 

3.9 Corollary. Let M be a ZG-module, and suppose that M is H-acyclic for 

each finite subgroup H of G; for example, this holds if M = AG, or if M is 

o(G)-inverting. Let X be an acyclic G-complex with finite stabilizers; for example, 

this holds if X is an EG. ThenH n (G, M) =0for all n>dim X. □ 

Here we can apply Theorem 2.6. 

3.10 Theorem. Let G be a locally finite group, and M a ZG-module which is 

H-acyclic for each finite subgroup H of G; for example, this holds if M = AG, or 

if M is o(G)-inverting. Then H n (G, M) =0for all n>ℵ-rank(G)+1. □ 

3.11 Remark. Theorem 3.10 can also be proved using the argument of the first 

paragraph of [11, Section 1]. □


4. Locally finite groups of cardinality ℵ 0 

In this section, we recall how H ∗ (G, AG) can be computed using an EG, and 

apply the method in the one-dimensional case. 

4.1 Definitions. Let X be an EG, or, more generally, any acyclic G-complex in 

which all cell stabilizers are finite, and let Notation 3.5 apply. 

We have natural identifications 

(4.2) A[[X]] = A X =Hom Z (ZX, A). 

For simplicity, let us suppose that X is finite dimensional. 

Then A[[X]] has the structure of a differential graded ZG-module, in which the 

differential ∂ ∗ has degree +1, and is given by 

∂ ∗ ( ∑ a x .x) = ∑ ( ∑ η(x, y)a y ).x. 

x∈X 

x∈X y∈X 

The cohomology of (A[[X]],∂ ∗ )isA concentrated in degree zero. 

Let 

A G [[X]] := { ∑ x∈X 

a x .x ∈ A[[X]] |{g ∈ G | a gx ≠0} is finite, for all x ∈ X}. 

Since G-stabilizers are finite, we see that A G [[X]] consists of all functions from 

X to A with finite support in each G-orbit. 

It is straightforward to check that A G [[X]] is a differential graded ZG-submodule 

of A[[X]]. 

We write C n (A G [[X]]), B n (A G [[X]]), and Z n (A G [[X]]) for the n-cochains, n-coboundaries, 

and n-cocycles, respectively. □ 

Sometimes the notation Hom c (Z[X],A)isusedtodenoteA G [[X]]; see, for example, 

[2, Lemma VIII.7.4]. 

The following is a variation on the usual “compact supports” cohomology; see, 

for example, [2, Proposition VIII.7.5]. It is particularly useful in the study of ends 

of groups. 

4.3 Theorem. If X is a finite-dimensional acyclic G-complex with finite stabilizers, 

then there is a natural isomorphism H ∗ (A G [[X]]) ≃ H ∗ (G, AG) of graded 

abelian groups. 

Proof. There is a natural identification of Hom ZG (ZX, A[[G]]) with Hom Z (ZX, A); 

see (3.2). There is also a natural identification of Hom Z (ZX, A) with A[[X]]; 

see (4.2). It is easy to show that under these identifications, Hom ZG (ZX, AG) 

corresponds to A G [[X]]. Hence H ∗ (A G [[X]]) ≃ H ∗ (Hom ZG (ZX, AG)). Finally, 

H ∗ (Hom ZG (ZX, AG)) ≃ H ∗ (G, AG), by Theorem 3.6. □ 

There is a natural right G-action on H ∗ (G, AG), arising from the ZG-bimodule 

structure on AG. This agrees with the natural right G-action on A G [[X]] which we 

have transformed into a left G-action. 

Let us illustrate how Theorem 4.3 can be used to study H ∗ (G, AG) whenG is 

locally finite of cardinality ℵ 0 . To do this we now construct a standardized EG, as 

in Remark 2.5.


4.4 Definition. Let G be a locally finite group of cardinality ℵ 0 . 

Index the elements of G with ω 0 ,soG = {h α | α


Hence φ(p) = 0, which is a contradiction. 

This proves that the composition 

A P → A[[Q]] → H 1 (A G [[X]]) = H 1 (G, AG) 

is injective. Since |P| = ℵ 0 , A ℵ 0 

embeds in H 1 (G, AG). □ 

5. Locally finite groups of cardinality ℵ 1 

In this section we study H ∗ (G, AG) whenG is locally finite with ℵ-rank(G) =1, 

topologizing and refining results of D. F. Holt. 

We begin by constructing a standardized EG. 

5.1 Definitions. Let G be a locally finite group of cardinality ℵ 1 . 

Let ω 1 ′ denote the set of limit ordinals less than ω 1. 

As in Definition 4.4, we start by indexing the elements of G, G = {h α | α


For α


Let Y be the subgraph of X obtained by deleting the vertices H and K and all 

their incident edges. It suffices to show that Y is a connected graph. Since X is 

connected, it suffices to show that each X-neighbour of H is Y -connected to each 

X-neighbour of K. Thus,leth ∈ H − K, andk ∈ K − H; it suffices to show that 

the vertices hK and kH are Y -connected. 

Let L = 〈hk −1 〉 = 〈kh −1 〉.SinceG is periodic, L is finite. 

We consider the action of L on X. Letm and n denote the orders of the L-orbits 

of the vertices H and K, respectively. By symmetry, we may assume that m ≥ n. 

Notice that L is not contained in K, son ≥ 2. 

Let g = kh −1 . In X, there is an edge H ∩ K joining H to K, and an edge 

k(H ∩ K) joining kH = gH to kK = K. Applying powers of g to these, we get a 

path in X with vertices 

H, K, gH, gK, g 2 H, g 2 K,... ,g n−1 H, g n−1 K. 

By the definition of m and n, these2n vertices are all distinct, so, on deleting the 

first two, we get a path in Y connecting gH = kH to g n−1 K = g −1 K = hK. □ 

5.3 Theorem (Holt [6]). If G is locally finite, and |G| = ℵ 1 ,thenH 1 (G, AG) =0. 

Proof. Let X be as in Definitions 5.1. 

Consider any φ ∈ Z 1 (A G [[X]]). Thus supp(φ) is a collection of edges of X, 

with only finitely many in each G-orbit. A subset of X which meets (that is, 

has nonempty intersection with) supp(φ) issaidtobebroken by φ. Since φ is a 

1-cocycle, we get 0 if we sum, in A, theφ-labels, with the appropriate signs, around 

any face, or along any closed path in the 1-skeleton, since X is simply connected. 

Thus there is a well-defined φ-sum from any vertex to any other vertex. 

Consider any α ∈ ω 1 ′ such that φ respects α as in the last paragraph of Definitions 

5.1. 

From Definitions 5.1, there is a cellular G α+1 -map M α → X, soφ induces an 

element φ α+1 ∈ A[[M α ]]. Since the G-stabilizers for X are finite, φ α+1 lies in 

A Gα+1 [[M α ]], Moreover, φ α+1 respects α in the obvious sense, since Y α is mapped 

to X α ,byconstruction. 

There exists n 0


H − ∩ K − . It follows that deleting g −1 K − p n from Y leaves a connected space 

containing G α+1,n0 v 1,0 . 

Suppose g ∉ G α+1,n0 ,sog ∈ G α+1,n −G α+1,n0 . Then supp(φ α+1 )∩gY ⊆ K − p n . 

Hence, on deleting supp(φ α+1 ) from the 1-skeleton of M α , one of the resulting 

components contains gG α+1,n0 v 1,0 . 

Next, we apply Lemma 5.2 to the graph Z ′ obtained by taking H = G α+1,n0 

and K = G α,n ,so〈H, K〉 = G α+1,n . We conclude that 

Z ′ − ({H}∪{K}∪(H ∪ K)/(H ∩ K)) 

is a connected graph. 

Let Z be the G α+1,n -subspace of M α generated by e · p n , and consider the map 

of G α+1,n -spaces from Z to Z ′ which assigns v 1,0 to H, v 0,n to K, ande · p n to 

H ∩ K. There is induced a surjective map 

Z − ({v 1,0 }∪{v 0,n }∪(H ∪ K)(e · p n )) → Z ′ − ({H}∪{K}∪(H ∪ K)/(H ∩ K)). 

Notice that φ α+1 breaks only edges of Z which lie in He ∪ Kp n ,sothereisa 

map from Z − ({v 1,0 }∪{v 0,n }∪(H ∪ K)(e · p n )) to the set of components of the 

1-skeleton of M α − supp(φ α+1 ). Moreover, we have seen that each subset gHv 1,0 

maps to a component of the 1-skeleton of M α − supp(φ α+1 ). Thus the map factors 

through Z ′ − ({H}∪{K}∪(H ∪ K)/(H ∩ K)), which is connected, so maps to a 

single component. Hence some component X ′ of the 1-skeleton of M α −supp(φ α+1 ) 

contains (〈H, K〉−H)v 1,0 ,thatis,(G α+1,n − G α+1,n0 )v 1,0 . 

Since n>n 0 was arbitrary, all of (G α+1 −G α+1,n0 )v 1,0 is contained in X ′ .Thus, 

for any path between any two elements of (G α+1 −G α+1,n0 )v 1,0 ,theφ α+1 -sum, and 

hence the φ-sum, is 0. 

Let ψ α+1 ∈ C 0 (A[[X α+1 ]]) be defined on each vertex v as the φ-sum along any 

path from any vertex of (G α+1 −G α+1,n0 )v 1,0 to v. Thenψ α+1 ∈ C 0 (A Gα+1 [[X α+1 ]]) 

and φ| Xα+1 = ∂ ∗ (ψ α+1 ). 

The α


We have a cellular G α -map Y α → X α , and, similarly, by Corollary 2.3, we can 

construct a cellular G 0 -map Y α • → Y 0 between spaces of type EG 0 .Thesetwomaps 

can be extended to a G α -map M α ′ → X α. 

Notice that Y α is contained in both M α and M α,andthemapY ′ α → X α has 

been extended to M α → X α+1 and to M α ′ → X α. 

For each n


Hence 

∂(h α,n (1 − g) 

n∑ 

f −1,i )=h α,n (1 − g)(x −1,0 + 

i=0 

n∑ 

(y 0,i − y −1,i )). 

We can view ψ α as an additive map Z[EM α ] → A, and apply it to the foregoing 

equation, to get 

φ α (h α,n (1 − g) 

n∑ 

f −1,i ) 

i=0 

=ψ α (h α,n (1 − g)(x −1,0 + 

i=0 

n∑ 

(y 0,i − y −1,i ))). 

Notice that h α,n G λα,n ∩ G λα,n = ∅. 

Since h α,n ,h α,n g ∉ G λ 

′ 

α,i 

,for0≤ i ≤ n, weseeφ α (h α,n (1 − g) n ∑ 

Also, h α,n ,h α,n g/∈ G µ ,soψ α (h α,n (1 − g)y −1,i )=0. 

Also, h α,n ,h α,n g/∈ G κ ,soψ α (h α,n (1 − g)x −1,0 )=0. 

∑ 

It follows that ψ α (h α,n (1 − g) n y 0,i )=0. 

i=0 

∑ 

But h α,n ,h α,n g/∈ G ν ,so(ψ α − ξ)(h α,n (1 − g) n y 0,i )=0. Hence 

ξ( 

i=0 

i=0 

n∑ 

h α,n (1 − g)y 0,i )=0. 

i=0 

i=0 

f −1,i )=0. 

Now ξ(h α,n y 0,n ) = −φ(p), and ξ vanishes on all other summands because 

h α,n G 0 ∩ G λα,i +1 = ∅ for 0 ≤ i ≤ n − 1, and gy 0,n ≠ y 0,n . Hence φ(p) =0, 

which is a contradiction. 

Since |P| = ℵ 1 , we have an embedding of A ℵ 1 

in H 2 (G, AG). □ 

6. Cohomology of directed unions 

In this section, we recall some known results about cohomology for well-ordered 

directed unions, with special emphasis on abelian groups. 

6.1 Notation. We let (P(G),∂) denote the bar resolution for G, andletP n (G) 

denote its nth component, for each n ∈ Z. Thus(P(G),∂)isafreeZG-resolution of 

Z, and, for n ≥ 0, P n (G) has as Z-basis the Cartesian power G n+1 ,withG acting 

by left multiplication on the first coordinate, and, for n ≥ 1, 

∂ n (g 0 ,... ,g n ):= 

n−1 

∑ 

(−1) i (g 0 ,... ,g i−1 ,g i g i+1 ,g i+2 ,... ,g n )+(−1) n (g 0 ,... ,g n−1 ). 

i=0 

As usual, if n ≤−1, P n (G) = 0, and, if n ≤ 0, ∂ n =0. 

The following is a degenerate case of the cohomology spectral sequence for 

well-ordered directed unions; see, for example [13, Section 3]. 

□


6.2 Lemma (Robinson [13, Proposition 1]). Let n ∈ N, letM be a ZG-module, 

let β be a limit ordinal, and let (G α | α ≤ β) be a continuous chain of subgroups of 

G. IfH n−1 (G α ,M)=0for all α


≠ 6.4 Theorem (Holt [6]). If G is locally finite, and ℵ-rank(G) 0, then 

H 1 (G, AG) =0. 

Proof. By (1.3), we may assume that ℵ-rank(G) ≥ 1, and we proceed by induction 

on ℵ-rank(G). If ℵ-rank(G) = 1, the assertion holds by Theorem 5.3. Thus we may 

assume that ℵ-rank(G) ≥ 2 and the result holds for smaller groups. There exists 

an ordinal β and a continuous chain (G α | α ≤ β) of subgroups of G with G β = G, 

and 1 ≤ℵ-rank(G α ) < ℵ-rank(G) for all α


6.8 Theorem. Let G be an abelian group, λ an ordinal, and ∆ a G-set with 

stabilizers of ℵ-rank strictly less than λ. 

(1) For each n ∈ N, ifℵ-rank(G) ≥ λ + n then H n (G, A∆) = 0. 

(2) If ℵ-rank(G) ≥ λ + ω 0 then H ∗ (G, A∆) = 0. 

Proof. (1). We argue by induction on n. 

If n =0,thenallG-stabilizers of elements of ∆ have infinite index in G, so∆ 

has no finite G-orbits. Thus (A∆) G =0,thatis,H 0 (G, A∆) = 0. 

Now suppose that n ≥ 1, and that the result holds for smaller n. Let β denote 

the least ordinal of cardinality rank(G), so β is a limit ordinal. Moreover, there 

exists a continuous chain of subgroups (G α | α ≤ β) such that G β = G, and, for 

each α


7. Cardinals, free abelian groups, and H F 

We now recall the hierarchies introduced in [9]; see [10] for more details. 

7.1 Notation. Let X denote a class of groups. 

All the classes of groups that we consider are closed under isomorphism, for 

example, the class F of all finite groups. 

We let L X denote the class of groups whose finitely generated subgroups all lie 

in X. For example, if X contains all finitely generated abelian groups, then L X 

contains all abelian groups. 

We let H1 X denote the class of all groups G for which there exists a finite-dimensional 

contractible G-complex with all stabilizers lying in X. For example, H1 F 

contains all finitely generated abelian groups, since, if G is finitely generated and 

abelian, then G has a finite subgroup N such that G/N is isomorphic to Z n for 

some n ∈ N, and thus G/N acts freely on R n preserving a CW-structure. 

If H1 X = X, thenX is said to be H1-closed. 

We let H X denote the smallest H1-closed class of groups which contains X. This 

class has a hierarchy indexed by the ordinals, where for each ordinal β, we define 

Hβ X recursively, by setting 

H0 X :=X, 

Hβ X :=H1 Hβ−1 X if β is a successor ordinal, 

Hβ X := ⋃ 

Hα X if β is a limit ordinal. □ 

α


7.3 Lemma. Let n ∈ N, letR be a ring, let 

0 → M n → M n−1 →···→M 1 → M 0 → M −1 → 0 

be an exact sequence of R-modules, and let L be an R-module. 

Suppose that Ext i R(L, M i )=0for i =0,... ,n.ThenExt 0 R(L, M −1 )=0. 

Proof. Clearly the result holds for n = 0. Thus we may assume that n ≥ 1, and 

that the result holds with n − 1inplaceofn. 

Let M ′ n−1 denote the cokernel of the map M n → M n−1 ,sowehaveexactsequences 

(7.4) 

(7.5) 

0 → M n → M n−1 →M n−1 ′ → 0 

0 →M n−1 ′ → M n−2 →···→M 0 → M −1 → 0 

Now (7.4) gives rise to a long exact sequence which contains the segment 

Ext n−1 

R 

(L, M n−1) → Ext n−1 

R 

(L, M n−1 ′ ) → Extn R (L, M n). 

Here the outer terms are zero, by hypothesis, so the inner term is zero. The 

induction hypothesis can now be applied to (7.5), and we see that Ext 0 R (L, M −1) = 

0. 

The result follows. □ 

We record the contrapositive of the case where R = ZG, andM −1 = L = Z with 

trivial G-action. 

7.6 Corollary. If n ∈ N, and 

0 → M n → M n−1 →···→M 1 → M 0 → Z → 0 

is an exact sequence of ZG-modules, then there exists i such that 0 ≤ i ≤ n and 

H i (G, M i ) ≠0. □ 

7.7 Proposition. If X is a class of groups, and H ∗ (G, Z∆) = 0 for every G-set ∆ 

for which all stabilizers lie in X, thenG ∉ H1 X. 

Proof. Suppose that G ∈ H1 X, so there exists a finite-dimensional, contractible 

CW-complex X on which G acts with all stabilizers lying in X. Let n denote the 

dimension of X. The augmented cellular chain complex of X, 

0 → C n (X) → C n−1 (X) →···→C 1 (X) → C 0 (X) → Z → 0, 

is an exact sequence of ZG-modules, so, by Corollary 7.6, there exists i such that 

0 ≤ i ≤ n and H i (G, C i (X)) ≠0. ThusH ∗ (G, C i (X)) ≠0. Butwecanwrite 

C i (X) =Z∆, where ∆ is the set of i-dimensional open cells of X, soisaG-set with 

all stabilizers in X. This contradicts the hypothesis. □ 

Combining Theorem 6.8(2) and Proposition 7.7, we get the following. 

7.8 Corollary. Let X be a class of groups and λ be an ordinal. Suppose that G is 

an abelian group in H1 X such that every subgroup of G which lies in X has ℵ-rank 

strictly less than λ. Thenℵ-rank(G)


7.9 Theorem. If β is any ordinal, then every abelian group in Hβ F has ℵ-rank 

strictly less than ω 0 β. 

Proof. We argue by induction on β. 

The result holds for β = 0 by definition of F. 

Thus we may assume that β>0, and that the result holds for smaller ordinals. 

Considerthecasewhereβ is a limit ordinal. Here, each abelian group in Hβ F 

lies in Hα F for some α


7.11 Remark. Theorem 7.10 gives an interesting new proof that 

H0 F ≠ H1 F ≠ H2 F ≠ H3 F. 

□ 

7.12 Conjecture. A ℵω0 +1 ∉ H 3 F; equivalently,A ℵω0 +1 ∉ H F. □ 

7.13 Conjecture. H3 F ≠ H F. □ 

7.14 Conjecture. There exists an ordinal α such that Hα F = H F. □ 

Acknowledgments. W. Dicks was partially supported by the DGES and the DGI 

through grants PB96-1152 and BFM2000-0354, respectively. I. Leary was partially 

supported by the EPSRC. S. Thomas was partially supported by NSF grants. 

I. Leary and P. Kropholler thank the Centre de Recerca Matemàtica of the 

Institut d’Estudis Catalans for the hospitality they received, and S. Thomas thanks 

the Departament de Matemàtiques of the Universitat Autònoma de Barcelona for 

its hospitality. 

References 

1. M. Auslander, On the dimension of modules and algebras, III, Nagoya Math. J. 9 (1955), 

67–77. 

2. Kenneth S. Brown, Cohomology of groups, GTM87, Springer-Verlag, Berlin, 1982. 

3. László Fuchs, Infinite abelian groups, Vol. I, Academic Press, New York, 1970. 

4. D. F. Holt, On the cohomology of locally finite groups, Quart.J.Math.Oxford32 (1981), 

165–172. 

5. D. F. Holt, The cohomological dimensions of locally finite groups, J. London Math. Soc. 24 

(1981), 129-134. 

6. D. F. Holt, Uncountable locally finite groups have one end, Bull. London Math. Soc. 13 (1981), 

557-560. 

7. T. Jech, Set theory, Academic Press, New York, 1978. 

8. P. H. Kropholler, Soluble groups of type FP ∞ have finite torsion-free rank, Bull. London 

Math. Soc. 25 (1993), 558–566. 

9. P. H. Kropholler, On groups of type FP ∞ ,J.PureAppl.Algebra90 (1993), 55–67. 

10. P. H. Kropholler and G. Mislin, Groups acting on finite dimensional spaces with finite stabilizers, 

Comment. Math. Helv. 73 (1998), 122–136. 

11. P. H. Kropholler and S. Thomas, The consistency of Holt’s conjectures on cohomological 

dimension of locally finite groups, J. London Math. Soc. 55 (1997), 76–86. 

12. B. Osofsky, Projective dimensions of “nice” directed unions, J. Pure Appl. Algebra 13 (1978), 

179–219. 

13. D. J. S. Robinson, Vanishing theorems for cohomology of locally nilpotent groups, pp. 120–129 

in: Group Theory, Proceedings, Brixen/Bressanone 1986, (Eds. O.H. Kegel, F. Mengazzo, 

G. Zacher), LNM 1281, Springer–Verlag, Berlin, 1987. 

14. Simon Thomas, An independence result in group cohomology, Bull. London Math. Soc. 28 

(1996), 264–268. 

15. Tammo tom Dieck, Transformation groups, Studies in Mathematics 8, de Gruyter, Berlin, 

1987. 

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classifying spaces for proper actions of locally finite groups

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